The "New" League of Legends Lore and my take on it. (my head canon)
Okay I'll preface this by saying I didn't search every nook and cranny of the boards and only am taking the "common knowledge" level of what players would know about the lore.
TL:DR The champions you play as are imitation puppets and this whole game is run for some gods' or mages' sick and twisted amusement.
The institute of war, summoners, etc. are all no longer canon (haven't been for years, really) and that's fine by me, but I've been trying to piece together an idea as to why the game exists, or really HOW it does.
Let me articulate; I've been thinking about it in the literal sense of it being a League of Legends. What the "League" is I don't know anymore really, but the way I see it is like this:
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The game as we know it you can say takes place at the very end of Runeterra's existence (present, even but we don't live there).
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Giant behemoths from beyond that devour everything and evolve, a madman with a long axe and a lizard, a little girl with a tragic past and unprecedented magical abilities. These are all quite literally legendary individuals. Even things as simple (subjectively) as kog'maw can be considered legends if they were something like a Boogeyman in the night or a mass murderer who grew to fame.
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Summoners technically exist in lore. Riot says they don't but I disagree. There's some group of collective Gods (higher than aurelion Sol and the Targonian aspects) or a group of really powerful mages that are pitting constructed and scaled avatars of legendary individuals of an age gone by just for fun (or salt depending on your view).
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They're scaled down just to equalize the playing field (in power, not necessarily, riot is working on that and has a team for it called "balance") for the sake of seeing something as baffling as Kled stabbing a star dragon made of pure celestial magic.
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Baron is an obvious legendary being, and the dragons probably were likely a family that were relevant somewhere in history. They were slain by some unknown group or individuals and are there because those same unnamed legends were gifted the powers we see in game.
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Most of the items have a name tied to them (Rabadon, Rylai, Sterak). Again these are artifacts of unimaginable power. They're in the game just to see what would happen if another legend got their hands on it. (The recipes and building other legendary items into even more powerful ones I can't explain away that's purely gameplay)
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New runes have some lore behind them and it's simply lifting their power and imposing it on the constructed avatars we play as.
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The lore we read did happen, and it's all canon. 90% of the interactions between champions is likely because they met in battle or as allies and the rest are Riot having a laugh for the sake of writing.
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Kayn is a weird exception. I can't really explain it well except as a "one did happen maybe but what if the other did" scenario.
I think I covered all my bases. I think we're still in a new lore relaunch four years in the making because some champions are still really outdated, but let me know what you think.
Edit: Sorry if there's weird words or spelling. Phone posting.